By a Jordan polygon P = *V*_{1} . . . *V*_{N} is meant a simple closed polygonal plane curve with *N* *sides* *V*_{l} *V*_{2}, *V*_{2} *V*_{3}, . . . , *V*_{N-1,} *V*_{N}, *V*_{N} *V*_{1}, joining the *N* *vertices* *V*_{1}, . . . ,*V*_{N}. In [**3**] any consecutive vertices *V*_{i-1}, *V*_{i}, and *V*_{i+1} of a Jordan polygon P are said to form an *ear* (regarded as the region enclosed by the triangle *V*_{i-1}* V*_{i }*V*_{i-1}) at the vertex *V*_{i} if the open chord joining *V*_{i-1 }and*V*_{i+1} lies entirely inside the polygon *P*. Two such ears are called *nonoverlapping* if the interiors of their triangular regions are disjoint. The following Two-Ears Theorem was proved in [**3**].

TWO-EARS THEOREM. *Except for triangles, every Jordan polygon has at least two* *nonoverlapping* *ears.*

The property of Jordan polygons expressed by the following theorem seems to provide a particularly simple and conceptual bridge from the Jordan Curve Theorem for Polygons to the Triangulation Theorem for Jordan Polygons; at least simpler perhaps than that given in Appendix B2 of [**5**].

Pairwise correlations are an important tool for understanding neuronal population activity [1]. Pairwise correlation graphs, where the edges reflect high levels of correlation between neurons, are often used as a proxy for underlying network connectivity. In this work, we are motivated by the question: *What is the structure of pairwise correlations in hippocampal population activity?* To address this question, we analyzed graded families of pairwise correlation graphs, parametrized by the threshold on correlation strength used to define the edges. In brain areas with receptive fields or place fields, the structure of the neural code has strong implications for structure of cliques in these graphs. This can be detected by examining the *clique topology* of the graph (specifically, topological invariants called ‘Betti numbers’ of the clique complex), and is closely tied to the dimension and topology of the underlying space [2]. For example, for hippocampal place cell activity during spatial exploration, pairwise correlation graphs are expected to have highly non-random and lowdimensional clique topology, due to the arrangement of place fields in a low-dimensional environment.

The purpose of this note is to provide a direct and relatively simple way of getting at the differentiability of

In the following, we will develop from the beginning the theory of repeating decimals. This is to provide the necessary machinery for the proof of Midy's theorem, as well as for completeness. ]]>

THEOREM.